Growth simulation for 3D surface and through-thickness cracks using ... · Growth simulation for 3D surface and through ... Symmetric Galerkin boundary element method; Finite

Journal of Mechanical Science and Technology 25 (9) (2011) 2335~2344

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0528-3

Growth simulation for 3D surface and through-thickness cracks

using SGBEM-FEM alternating method† Jai Hak Park1,* and Gennadiy P. Nikishkov2

1Department of Safety Engineering, Chungbuk National University, Chungbuk 361-763, Korea 2Department of Computer Science and Engineering, The University of Aizu, Fukushima 965-8580, Japan

(Manuscript Received August 23, 2010; Revised April 6, 2011; Accepted April 29, 2011)

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract An SGBEM-FEM alternating method had been proposed by Nikishkov, Park and Atluri for the analysis of three-dimensional planar

and non-planar cracks and their growth. The proposed method is an effective method for fatigue or stress corrosion crack growth simula-tion. During crack growth simulation, however, an oscillation phenomenon is observed in crack advance or stress intensity factor distri-bution. If oscillating amplitude in SIF or crack advance does not decrease during next increment steps, the crack growth simulation fails. In this paper several methods are examined to remove the oscillation phenomenon. As a result, it is found that smoothing in stress inten-sity factor distribution or in crack front geometry can remove or weaken the oscillation phenomenon. Using the smoothing techniques, stress corrosion crack growth simulation is performed for a semi-elliptical surface crack and a through-thickness crack embedded in a plate. Crack front shape and stress intensity factor distribution are obtained after each increment during the crack growth. And the depth and length of a crack are obtained as a function of time. It is noted that the SGBEM-FEM alternating method is a very effective method for SCC growth simulation for a surface crack and a through-thickness crack.

Keywords: Symmetric Galerkin boundary element method; Finite element method; Alternating method; Stress corrosion cracking (SCC); Three-

dimensional crack ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction

For several decades, the Shwartz-Neumann alternating tech-nique has been developed for three-dimensional cracks [1-4]. Nikishkov, Park and Atluri [5] proposed an SGBEM-FEM alternating method to analyze planar or non-planar three-dimensional cracks in a finite body. They used the symmetric Galerkin boundary element method (SGBEM) [6, 7] for mod-eling a crack embedded in an infinite body.

To perform fatigue or stress corrosion crack growth simula-tion, fracture parameters such as stress intensity factor (SIF) should be obtained along the crack front during crack growth. The well-established finite element method can be used for the simulation, but it is difficult to model growing cracks due to complications related to generation and modifications of finite element mesh during crack growth. The proposed SGBEM-FEM alternating method is a convenient method to perform crack growth simulation. Since the boundary element mesh (crack mesh) is independent of the finite element mesh, crack growth can be simulated by just changing the boundary ele-

ment mesh. During crack growth simulation, however, an oscillation phenomenon is observed in crack advance and stress intensity factor distribution. The phenomenon occurs for the following reason. If a crack front point advances less than adjacent crack front points due to calculation error or local geometry, the SIF of the point becomes larger than the values of adjacent crack front points. So in the next increment, the crack front point advances more than other adjacent points. Then after the increment, the SIF of the point becomes less than other points. If oscillating amplitude in SIF or crack ad-vance does not decrease during next increment steps, the crack growth simulation fails. Fig. 1 shows typical oscillation phe-nomenon in SIF distribution of a growing surface crack.

We propose a methodology that removes the oscillation phenomenon in crack growth simulation. As a result, it is found that geometrical smoothing of crack front and smooth-ing in SIF distribution can remove or weaken the oscillation phenomenon. Starting from a small initial surface crack growth simulation is made until the crack becomes a large surface crack using the smoothing method.

If a growing surface crack satisfies the crack growth insta-bility criterion, unstable crack growth occurs and the crack becomes a through-thickness crack. Since a through-thickness crack also grows due to stress corrosion, it will be convenient

† This paper was recommended for publication in revised form by Associate EditorChongdu Cho

*Corresponding author. Tel.: +82 43 261 2460, Fax.: +82 43 264 2460 E-mail address: [email protected]

© KSME & Springer 2011

2336 J. H. Park and G. P. Nikishkov / Journal of Mechanical Science and Technology 25 (9) (2011) 2335~2344

to include SCC growth of a through-thickness crack for a complete crack growth simulation. Park, Kim and Nikishkov [8] examined the applicability of the SGBEM-FEM alternat-ing method to a through-thickness and found that accurate SIF can be obtained using the method. Starting from a small sur-face crack, SCC growth simulation is performed including the later stage of the growth of a through-thickness crack. From the growth simulation, crack shape and stress intensity factor distribution are obtained after each increment during the crack growth. The crack depth and crack length are obtained as a function of time.

2. Analysis method

2.1 SGBEM-FEM alternating method

In this paper, the modeling of SCC crack growth is per-formed using the combination of the finite element method (FEM) and the symmetric Galerkin boundary element method (SGBEM) [6, 9, 10]. The uncracked structural component is discretized with finite elements. The crack is modeled with the SGBEM. The alternating method [5, 11] is used to combine both methods using the superposition principle. The equilib-rium state for the system of the structural component (FEM) and the crack (SGBEM) is reached as a result of iterations that alternate between two methods. The crack is represented by a set of quadratic eight-node boundary elements. Singular boundary elements are used at the crack front. Fig. 2 shows typical crack mesh for a semi-elliptical surface crack with the crack depth a and the crack length 2C. The lower mesh with the length Lext is located outside the body. The stress intensity factors KI, KII and KIII at the crack front are calculated through displacements of nodes near the crack front.

A crack growth procedure is based on the effective stress in-tensity factor. It is supposed that the crack grows in the direc-tion of the J-integral and the crack growth rate is determined by the value of the effective stress intensity factor Keff. The value of Keff is calculated through the J-integral value. The crack front advancement is performed by adding a new boun-dary element layer to the existing crack model. Crack growth rate in the case of SCC is determined with an empirical model [12, 13] using the effective stress intensity factor.

The developed crack growth procedure for three-dimensional mixed mode cracks is implemented in the Java SGBEM-FEM code. Examples of non-planar growth of semi-elliptical cracks under SCC conditions are presented.

The SGBEM equilibrium equation system for a crack in an infinite medium is written in the following matrix form:

[ ]{ } { }.BEM BEMK U T= (1) Here [ ]BEMK is the global SGBEM matrix, { }BEMU is the

global displacement discontinuity vector and { }T is the nodal equivalent of the crack surface forces. Eight-node quad-ratic boundary elements are employed for crack modeling. Elements can have curved edges and consequently, a curved surface. This allows representing arbitrary three-dimensional cracks with non-planar surface.

Solution of the boundary value problem for a structural component with a crack is sought as superposition of the finite element solution (uncracked finite body) and the boundary element solution (crack in an infinite medium). For a correct superposition, fictitious forces on the boundary of the finite element model should be found in order to compensate for the stresses caused by the presence of a crack in an infinite body. These fictitious forces can be efficiently found with the alter-nating procedure:

1

1

(0) (0) (0)

( ) ( 1)

( ) ( 1) ( )

( ) ( )

( ) ( )

( ) ( )

(

{ } {0}, { } { }, { } {0}do iterations

{ } [ ] { }

{ } { } { }

{ } [ ][ ]{ }

{ } [ ][ ]{ }

{ } [ ] { }

{

FEM BEM

i iFEM FEM

i i iFEM FEM FEM

i iFEM FEM

i iBEM FEM

S

i iBEM BEM

iBEM

U P U

U K

U U U

E B U

T N n dS

U K T

U

σ

σ

−

−

−

−

= Ψ = =

Δ = Ψ

= + Δ

Δ = Δ

Δ = Δ

Δ = Δ

∫

) ( 1) ( )

( ) ( )

( ) ( )

( )

} { } { }

{ } { ({ })}

{ } [ ][ ]{ }

until || || / || || .

i iBEM BEM

i iBEM BEM

i iFEM BEM

S

i

U U

U

N n dS

P

σ σ

σ

ε

−= + Δ

Δ = Δ Δ

Ψ = Δ

Ψ <

∫

(2)

Equilibrium superposition is determined as a result of itera-

tions. Superscripts in parentheses denote iteration numbers. During iterations global matrices [ ]FEMK and [ ]BEMK do not change. In the above relations, { }Ψ is the residual force vec-

Fig. 1. Typical oscillation phenomenon in SIF distribution.

Fig. 2. Typical crack mesh.

J. H. Park and G. P. Nikishkov / Journal of Mechanical Science and Technology 25 (9) (2011) 2335~2344 2337

tor, { }σ is the stress vector, [ ]B is the finite element dis-placement differentiation matrix, and [ ]E is the elasticity matrix. The iterative loop ends when the relative norm of the finite element residual becomes less than the specified error tolerance .ε After termination of the iterative procedure, correct tractions at the crack surface are determined thus mak-ing possible to estimate values of the stress intensity factors at the crack front.

2.2 Calculation of fracture mechanics parameters

Once the displacement discontinuities are obtained at nodes, the stress intensity factors KI, KII and KIII can be easily deter-mined using the following relations [14]:

32

22

1

(1 ) 4 2

(1 ) 4 2

(1 ) 4 2

I

II

III

E uKr

E uKr

E uKr

πν

πν

πν

=−

=−

=+

(3)

where E is the elasticity modulus; ν is Poisson’s ratio; r is the distance from the point to the crack front; and u1, u2 and u3 are components of the displacement discontinuities at points on the crack surface in a local crack front coordinate system, x1, x2 and x3, which are illustrated in Fig. 3. The axis x1 of the crack front coordinate system is parallel to the crack front, and the axis x3 is normal to the crack surface.

The following procedure for the stress intensity factor cal-culation is used in the current work:

(1) Obtain the displacement discontinuities Giu in the global

coordinate system for the quarter-point node and for the cor-ner node of a singular crack front element.

(2) Extrapolate Giu r to the crack front, using values at

the quarter-point node (L/4) and at the corner node (L). Here r is the distance along the line normal to the crack front and

Giu components of displacement discontinuities in the global

coordinate system. (3) Transform the extrapolated displacement discontinuities

from the global coordinate system to the crack front coordi-nate system, G

i ij iu uα= where ijα are direction cosines of the transformation.

(4) Calculate the stress intensity factors using Eq. (3).

3. Modeling crack growth

3.1 Modeling of non-planar crack growth

The SGBEM-FEM alternating method is quite suitable for crack growth simulation. Since the crack is modeled sepa-rately, the finite element model need not be modified during crack growth. Only the boundary element model (crack model) should be changed during crack growth. For crack growth simulation of a non-planar crack, it is necessary to know the direction of crack growth and the amount of crack growth. The J-integral is used to determine the crack growth direction and the amount of crack growth as follows [8]:

(1) Crack grows in the direction of J-integral vector as shown in Fig. 4;

(2) Crack growth rate is determined by the effective stress intensity factor effK based on the J-integral.

In an elastic three-dimensional case, the J-integral compo-nents are evaluated using the stress intensity factors as:

2

2 2 21

2

2

2 21 2

1 1( )

12

.

I II III

I II

J K K KE E

J K KE

J J J

ν ν

ν

− += + +

−= −

= +

(4)

The crack growth angle α, which is the angle between the

axis x1 and the crack growth direction, is determined by the direction of J-integral vector:

2

1

tan .JJ

α = (5)

Fig. 3. The global coordinate system X1, X2, X3 and the crack frontcoordinate system x1, x2 and x3.

J2

x2

αx1

x3J1

J

Fig. 4. Components J1 and J2 of the J-integral at the crack front. It is supposed that the crack grows in the direction of the J-integral.


It is worth noting that the J-integral vector is normal to the crack front. Hence, a point at the crack front moves in the plane normal to the crack front at the angle α, from the plane which is tangential to the crack surface.

A typical crack growth model, suitable for fatigue or SCC crack growth simulation can be expressed using the effective stress intensity factor Keff as follows:

( )effda f Kdt

= (6)

where da/dt is the crack growth rate and Keff is related to the J-integral as:

2 .1eff

JEKν

=−

(7)

3.2 Crack growth algorithm The following algorithm is used to model mixed mode SCC

crack growth for a surface crack [8]: (1) Solve the boundary value problem for the current crack

configuration using the SGBEM-FEM alternating method. (2) Obtain the stress intensity factors KI, KII and KIII for the

element corner nodes located at the crack front and calculate the effective stress intensity factor effK according to Eq. (7) and select the maximum value max

effK . (3) Estimate increment of the crack life by the following in-

tegration and accumulate the crack life t t t= + Δ :

.( ( ))

a a

effa

datf K a

+Δ

Δ = ∫ (8)

(4) If no value of crack advance Δamax is left in the input

data, then stop. (5) For each corner node, determine the crack front coordi-

nate system by averaging the coordinate axis vectors deter-mined at the corner point of two neighboring boundary ele-ments. Also determine x1, x2 and x3 local coordinate system illustrated in Fig. 4.

(6) For each corner node, calculate the crack growth angle α according to Eq. (5).

(7) Determine crack advance Δa for the corner nodes at the crack front using the following equation (see Fig. 5):

max max

( )( )

eff

eff

f Ka a

f KΔ = Δ (9)

(8) Move the corner nodes along the J-integral vector ac-

cording to computed Δa values. (9) Transform the coordinates of the neighbor nodes into the

x1, x2 and x3 coordinate system. Perform curve fitting for x1 and x2 coordinates with respect to x3 coordinate using a poly-nomial expression.

(10) Using the fitted polynomial coefficients, obtain the x1 and x2 coordinates of the advance point. For the crack front points located outside the body, we use the same Δa and α as the values calculated at the nearest crack front point on the body boundary.

(11) Find the locations of crack front midside nodes, using linear or cubic spline interpolation.

(12) Shift the quarter-point nodes of the previous crack front elements to midside position. Put quarter-point nodes on element sides nearly normal to the crack front.

(13) Generate one layer of boundary elements between old and new crack fronts.

(14) Go to step (1).

3.3 Geometrical smoothing of crack front

When crack growth simulation is performed using the alter-nating method, the oscillation phenomenon is observed in crack advance and SIF distributions as illustrated in Fig. 1. To eliminate the oscillation phenomenon, geometrical smoothing of crack front can be used. Fig. 6 illustrates the procedure of geometrical smoothing.

Consider a procedure to find a new advancing crack front point A′ corresponding to the current crack front point A. First obtain the crack advance points corresponding to the current crack front points. The open circles in Fig. 6 denote the crack advance points. Next, transform the coordinates of the crack advance points into the x1, x2 and x3 coordinate sys-tem in Fig. 4, which is the local coordinate system at A. Note that the local coordinate system changes according to the cur-rent crack front point. Curve fitting is performed for the x1 and x2 coordinates with respect to x3 coordinate using a polynomial expression. Let the number of points used in the curve fitting be nfit. An even nfit value can remove the oscillating phenome-

ΔamaxΔa

Keffmax Keff

Fig. 5. Advancement of the crack front. Points at the crack fronts are moved in the directions of the J-vector. Maximum crack advance

max .aΔ

1x 3x

Fig. 6. Geometrical crack front smoothing.


non more effectively than an odd nfit value. In this study, the second order polynomial is used in the curve fitting and nfit = 6 is used. Using the fitted polynomials, the x1 and x2 coordinates of A′ can be calculated easily. This procedure can be applied to planar and non-planar crack growth.

Normally, the SIF value on the body boundary is less than the value inside the body. If we use the SIF values on the boundary in crack growth simulation without modification, it may be the source of the oscillation phenomenon. Instead of the obtained SIF value, an extrapolated SIF value using inner 2 or 3 points can be used. It is also possible to perform crack front geometrical smoothing excluding the front points on the body boundary. In this case, the obtained SIF value on the body boundary is not used.

3.4 Smoothing of stress intensity factor distribution

Smoothing of the SIF distribution can be used instead of geometrical smoothing technique. After calculating SIF values for crack front corner nodes, we can modify the values through smoothing technique. To obtain a modified SIF value, curve fitting is performed for modes I, II and III SIF values. Consider a crack front point A, at which a modified SIF value is desired. First, choose nfit points including the point A and the nearest adjacent crack front corner points. Transform the coordinates of the chosen points into the x1, x2 and x3 local coordinate system of the point A. Perform curve fitting for the SIF values of the points with respect to the local x3 coordinate using a polynomial expression. Using the obtained polynomi-als, the mixed mode SIF values at point A can be obtained easily. As in the geometrical smoothing, an even nfit value can remove the oscillating phenomena more effectively than an odd nfit value.

3.5 Crack front shifting

In the current technique, one crack front element layer is added to simulate the increase in crack size. The maximum increment can be controlled by using the input data. But if the SIF values for a part of crack front are much less than the maximum SIF value, a crack front element with short edge is generated and the simulation fails because of the poor element quality. To prevent generating an element with poor quality, crack shifting can be used as shown in Fig. 7. In the method, the crack front nodes are moved to the new position without generating a new crack front element layer. In Fig. 7(a), the open circles denote the new positions of crack front points. This technique also curbs the increase in number of boundary elements and eventually reduces the simulation time.

When a crack front element with short edge is generated, the element quality can be improved by adjusting the element heights of two current crack front element layers as shown in Fig. 8. After generating a new crack front element layer, the coordinates of the previous crack front points are modified so that the two current element layers have nearly the same width. If this technique is used in conjunction with crack shifting, a

small increment becomes possible in the crack growth simula-tion.

3.6 Crack growth material model

As an example, consider crack growth under SCC condi-tions. Several material models for determining the SCC crack growth rate in the stainless steel-water systems have been developed [12, 13]. Currently for testing the developed crack growth procedure, we use the mechanochemical model pro-posed by Saito and Kuniya [12]. The model is represented by the following equation:

( )( )2 /( 1)

0 1 2 3 4exp ( ) .mn

eff ISCCda A C C C C K Kdt

+⎡ ⎤= − − −⎢ ⎥⎣ ⎦ (10)

(a)

(b)

Fig. 7. Crack front shifting technique.

(a)

(b)

Fig. 8. Adjustment of the heights of two current crack front elements.


Here, Keff is the effective stress intensity factor calculated through the J-integral value, A0, C1, C2, C3 and C4 are material constants, KISCC is the threshold stress intensity factor, n is the Ramberg-Osgood type strain hardening coefficient, m is the parameter representing the effect of environment and material chemistry.

3.7 Through-thickness crack

The initiated crack due to SCC is very small, but it can grow to a large surface crack with time. If a growing crack satisfies an instability criterion it becomes a through-thickness crack. For a long through-thickness crack two-dimensional SIF solutions can be used to simulate crack growth. But if the crack length is short or the crack front is not straight or normal to the surface, a three-dimensional SIF solution could be used. For this purpose, the SGBEM-FEM alternating method is adequate. The accuracy of SGBEM-FEM alternating method is fully examined for a surface crack [5, 15] and for a through-thickness crack [8] in other references. Park, Kim and Nikish-kov [8] verified that accurate SIF values can be obtained and crack growth simulation can be performed for a through-thickness crack using SGBEM-FEM alternating method. The crack growth algorithm for a through-thickness crack is also given in the reference.

4. SCC growth simulation for a semi-elliptical surface

crack SCC growth simulation is performed for a plate with an ini-

tial inclined semi-elliptical surface crack. A plate has thick-ness t, width 2W and height 2h as illustrated in Fig. 9. It is subjected to tension with surface intensity. Initial semi-elliptical crack with aspect ratio a/c is located at the center of the specimen surface and oriented under angle β to the hori-zontal plane. Plate and loading parameters have the following values: t = 0.2 m, 2W = 0.64 m, 2h = 0.6 m, σ = 200 MPa. And crack parameters are: a = 0.01 m, a/c = 0.5, β = 0, 15,

45°. The body is assumed as an elastic material with elastic modulus E = 210 GPa and Poisson’s ratio ν =0.3. The SCC Saito-Kuniya model Eq. (10) is used for prediction of SCC crack growth with the following parameters: A0=1.1 × 10-7, C1=2.5 × 1010, C2=12.9199, C3=3.0, C4=0.15, KISCC=9.0 MPa m1/2 and n=5. These values are those for type 304 stainless steel in water at 288°C and the unit of da/dt is m/s in Eq. (10).

Fig. 10 shows an example of finite element mesh and boundary element mesh (crack mesh). Only a half of the finite element mesh is plotted. In the finite element mesh, 4,907 nodes and 980 20-node three-dimensional solid elements are used. Fig. 11 shows boundary element mesh for an initial crack. The boundary element mesh consists of 257 nodes and 72 8-node boundary elements. To represent stress singularity at the crack front, the midside nodes are moved to the quarter positions in crack front elements. In Fig. 11 the lower element layer with the height of Lext is the fictitious portion of the boundary element mesh and located outside the body. The fictitious portion improves the accuracy of SIF solution. The run time is 73 sec for one increment on an Intel 3 MHz per-sonal computer for the typical model in Fig. 10. The run time is strongly dependent on the size of SGBEM model. The specified damax value has an effect on the accuracy and stabil-ity of the simulation. If an inaccurate SIF value is obtained, it is necessary to adjust the damax value, or to use the crack front shifting technique.

2Wt

a

σ

2h

ϕ

2c

β

(a) (b) Fig. 9. Inclined semi-elliptical surface crack in a plate subjected to a uniform tensile stress. Location of points on the crack front is charac-terized by elliptical angle ϕ .

Fig. 10. Example of finite element mesh and initial boundary element mesh (crack mesh) used in the study. Only half of the finite element mesh is plotted.

Fig. 11. Initial boundary element mesh (crack mesh).


4.1 Planar crack growth First planar crack growth is considered. The problem is the

case when β = 0° in Fig. 9. Fifteen crack advances are per-formed with specified damax values, whose values are 0.003 m, 0.004 m, 0.005 m × 10 and 0.006 m × 3. Fig. 12 shows the Mode I SIF distributions during the crack growth when the geometrical smoothing is used. The SIF is normalized by K0, where K0 is [14]:

0aK

Qπσ= (11)

1.65

1 1.464 .aQc

⎛ ⎞= + ⎜ ⎟⎝ ⎠

(12)

After each increment the normalized SIF is plotted as a

function of angle θ . Angle θ is defined by arctan( / )y xθ = . For the initial crack, the SIF values at the deep points are greater than the values at the points near surface. As the crack grows, the SIF values become nearly constant excluding the values on the surface. During the crack growth, Mode II and Mode III SIF maintain very small value. The maximum Mode II or Mode III SIF is less than 1% of Mode I SIF.

Fig. 13 shows the variation of Δa/a and Δc/a as a function of time. Δa and Δc are the total increments in the crack depth and the half crack length respectively. In the initial stage the growth rate of Δa/a is larger than that of Δc/a. As time goes on, the two growth rates become equal. Fig. 14 shows the final

boundary element mesh after 15 increments. Fig. 15 shows the Mode I SIF distributions during the crack

growth when SIF smoothing is used. Each curve represents SIF distribution before smoothing after each increment. So each SIF distribution becomes smoother after the smoothing procedure. On the boundary, the SIF values obtained by linear extrapolation using two internal points are used in crack growth simulation.

4.2 Non-planar crack growth

Next consider SCC growth simulation for an inclined initial surface crack with β = 15° in Fig. 9. Fifteen crack advances are performed with specified damax values, whose values are 0.003 m, 0.004 m, 0.005 m× 8, 0.007 m × 2 and 0.009 m× 3. Fig. 16 shows normalized effective SIF, Keff/K0 distributions

Fig. 12. Normalized SIF distributions during the crack growth for aninitial crack with β =0. The geometrical smoothing technique is used.

Fig. 13. Normalized increments of crack depth and half crack length asa function of time. The geometrical smoothing is used.

Fig. 14. Boundary element mesh after 15 increments for a crack with β= 0°.

Fig. 15. Normalized SIF distributions during the crack growth for an initial crack with β =0°. The SIF smoothing technique is used.

Fig. 16. Normalized effective SIF distributions during the crack growth for an initial crack with β =15°. The geometrical smoothing technique is used.


during the crack growth. Here, Keff is calculated using Eq. (7) and K0 is defined by Eq. (11). In the initial stage of growth the SIF at deep points are larger than those of the points near the surface. But as the crack grows, the SIF values become equal at all points except the points on or just near the surface. Nor-malized Mode II and Mode III SIF distributions during the first three increments are given in Fig. 17. Before growth, the maximum normalized Mode II SIF is 0.19. During the next three increments, the values become -0.09, 0.034 and -0.028. It can be noted that the magnitude of Mode II SIF decreases as the crack grows. Mode III SIF also decreases as the crack grows, but the decreasing rate is small. Fig. 18 shows the final boundary element mesh after 15 increments. Initially inclined crack surface changes to almost a flat surface which is normal to the loading direction as the crack grows.

Figs. 19 and 20 show normalized effective and Mode I SIF distributions during the crack growth for an initial inclined surface crack with β = 45°. The specified damax values are the same as in the case when β = 15°. Fig. 21 shows normalized Mode II and Mode III SIF distributions during the first three increments. The magnitude of Mode II SIF decreases rapidly as the crack grows, but Mode III SIF maintains nearly the same value during the first three increments. According to the results, the Mode III SIF decreases to a small value very slow-ly as the crack grows. Fig. 22 shows the final boundary ele-ment mesh after 15 increments.

Fig. 17. Normalized Mode II and III SIF distributions during the crack growth for an initial crack with β =15°. The geometrical smoothing technique is used.


Fig. 19. Normalized effective SIF distributions during the crack growth for an initial crack with β =45°. The geometrical smoothing technique is used.

Fig. 20. Normalized Mode I SIF distributions during the crack growth for an initial crack with β =45°. The geometrical smoothing technique is used.

Fig. 21. Normalized Mode II and III SIF distributions during the crack growth for an initial crack with β =45°. The geometrical smoothing technique is used.



5. SCC growth simulation from a surface crack to a through-thickness crack If a growing surface crack satisfies an instability criterion,

unstable crack growth occurs and it becomes a through-thickness crack. For complete SCC growth simulation the growth of a through-thickness crack should be included. Start-ing from a small semi-elliptical surface crack SCC growth simulation is performed until the crack becomes a through-thickness crack with moderate length.

Net section yielding or tearing modulus can be used as an instability criterion [14]. But in this study, it is assumed that if the crack depth reaches 80% of the thickness it becomes a through-thickness crack. Fig. 23 shows an initial shape of a through-thickness with unequal surface lengths 2a1 and 2a2. For a conservative analysis, we can use the assumption, a2 = a1 = cf for an initial through-thickness. Here cf is the half crack length of the surface crack when the instability criterion is satisfied. In this study we assume that a1 = cf, a2 = 0.5 a1.

A plate has thickness t, width 2W and height 2h. It is sub-jected to tension with surface intensity σ. Plate and loading parameters have the following values: t = 0.05 m, 2W = 0.64 m, 2h = 0.6 m, σ = 200 MPa. And the initial crack parameters are: a = 0.01 m, a/c = 0.5. The finite element mesh and bound-ary element mesh for a surface crack are the same as in Figs. 10 and 11. Fig. 24 shows finite element mesh and boundary element mesh (crack mesh) for a through-thickness crack.

Only a half of the finite element mesh is plotted. In the finite element model, 4,907 nodes and 980 20-node three-dimensional solid elements are used and in the crack mesh, 789 nodes and 240 8-node boundary elements are used.

Fig. 25 shows the final boundary element mesh when the crack depth of the surface crack reaches 80% of the thickness. The final half crack length cf is 0.0451 m. Fig. 26 shows the final boundary element mesh for the through-thickness crack. Only the mesh inside of the body is plotted. The shaded area represents the boundary element mesh for the initial through-thickness crack. Fig. 27 shows variation of normalized crack size as a function of time. For the crack size parameter, crack depth is chosen for a surface crack and half crack length at the middle of the plate is chosen for a through-thickness crack. The crack size is normalized by the thickness t. The disconti-nuity in the curve is because the crack is changed from a sur-face crack to a through-thickness crack. It is noted that the crack growth rate gradually increases with time.

6. Conclusion

Crack growth simulation is performed using the SGBEM-FEM alternating method. It is found that the oscillation phe-

Fig. 23. SCC growth model used in this study.

Fig. 24. FEM mesh and boundary element mesh (crack mesh) for athrough-thickness crack. Only a half of the finite element mesh isplotted.

Fig. 25. Final boundary element mesh for a growing surface crack.

Fig. 26. Final boundary element mesh for a growing through-thickness crack.

Fig. 27. Variation of normalized crack size as a function of time.


nomenon can be removed using geometrical smoothing of crack front or smoothing of stress intensity factor distribution. For complete SCC growth simulation, the growth of a through-thickness crack is also considered. Starting from a small initial surface crack, crack growth simulation is made in a stainless steel plate until the crack becomes a large surface crack or a through-thickness crack. Crack shape and stress intensity factor distribution are obtained after each increment during the crack growth. The crack depth and crack length are obtained as a function of time. It is found that the SGBEM-FEM alternating method can be used as an effective method to analyze SCC growth simulation for a surface crack and a short through-thickness crack.

References

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[4] Z. D. Han and S. N. Atluri, SGBEM (for cracked local sub-domain)-FEM (for uncracked global structure) alternating method for analyzing 3D surface cracks and their fatigue-growth, Computer Modeling in Engineering & Sciences, 3 (6) (2002) 699-716.

[5] G. P. Nikishkov, J. H. Park and S. N. Atluri, SGBEM-FEM alternating method for analyzing 3D non-planar cracks and their growth in structural components, Comp. Modeling in Engng & Sci., 2 (3) (2001) 401-422.

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[8] J. H. Park, M. W. Kim and G. P. Nikishkov, SGBEM-FEM alternating method for simulating 3D through-thickness crack growth, Computer Modeling in Engineering & Sci-ences, 68 (2010) 269-296.

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Jai Hak Park received his M.S. and Ph.D. in Mechanical Engineering from KAIST. He is currently a professor at Chungbuk National University. His research interests are in the area of frac-ture mechanics, computational mechan-ics and probabilistic assessment of structure.

Gennadiy Nikishkov received his Ph.D. and D.Sc. in Computational Mechanics from the Moscow Engineering Physics Institute. He held a Professor position at the Moscow Engineering Physics Insti-tute. Dr. Nikishkov is currently a Profes-sor at the University of Aizu, Japan. His research interests include computational

modeling, high performance computing, visualization and computer graphics.

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Growth simulation for 3D surface and through-thickness cracks using ... · Growth simulation for 3D surface and through ... Symmetric Galerkin boundary element method; Finite